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Document from the year 2020 in the subject Mathematics - Stochastics, Sardar Patel University, language: English, abstract: This Lecture Notes teach the advancement from basic probability theory to central limit theorem as a rigorous mathematical subject. Chapter 1 provides an introduction to random variables, probability space and distribution functions. Chapter 2 presents joint distribution functions, expectations and some well known inequalities concern with the expectation. Chapter 3 concern with the various types of convergence of random variables and related theorems. Also strong law of large numbers has been discussed. Chapter 4 deals with characteristic functions, inversion formula and weak convergence. Sufficient number of examples are given to digest theory.